COMPUTER PROCESSING AND MODELING Full Papers
Magnetic Resonance in Medicine 76:645–662 (2016)
Magnetic Resonance Elastography of the Brain: An In Silico Study to Determine the Influence of Cranial Anatomy Deirdre M. McGrath,1,2* Nishant Ravikumar,3 Iain D. Wilkinson,2,4 Alejandro F. Frangi,1,4y and Zeike A. Taylor3,4y Purpose: Magnetic resonance elastography (MRE) of the brain has demonstrated potential as a biomarker of neurodegenerative disease such as dementia but requires further evaluation. Cranial anatomical features such as the falx cerebri and tentorium cerebelli membranes may influence MRE measurements through wave reflection and interference and tissue heterogeneity at their boundaries. We sought to determine the influence of these effects via simulation. Methods: MRE-associated mechanical stimulation of the brain was simulated using steady state harmonic finite element analysis. Simulations of geometrical models and anthropomorphic brain models derived from anatomical MRI data of healthy individuals were compared. Constitutive parameters were taken from MRE measurements for healthy brain. Viscoelastic moduli were reconstructed from the simulated displacement fields and compared with ground truth. Results: Interference patterns from reflections and heterogeneity resulted in artifacts in the reconstructions of viscoelastic moduli. Artifacts typically occurred in the vicinity of boundaries between different tissues within the cranium, with a magnitude of 10%–20%. Conclusion: Given that MRE studies for neurodegenerative disease have reported only marginal variations in brain elasticity between controls and patients (e.g., 7% for Alzheimer’s disease), the predicted errors are a potential confound to the development of MRE as a biomarker of dementia and other neurodegenerative diseases. Magn Reson Med 76:645–662, C 2015 Wiley Periodicals, Inc. 2016. V
Key words: magnetic resonance elastography; simulation; finite element modeling; brain; dementia; Alzheimer’s disease
INTRODUCTION
1 CISTIB Centre for Computational Imaging & Simulation Technologies in Biomedicine, Department of Electronic and Electrical Engineering, The University of Sheffield, Sheffield, UK. 2 Academic Unit of Radiology, Faculty of Medicine, Dentistry & Health, The University of Sheffield, Sheffield, UK. 3 CISTIB Centre for Computational Imaging & Simulation Technologies in Biomedicine, Department of Mechanical Engineering, The University of Sheffield, Sheffield, UK. 4 INSIGNEO Institute for In Silico Medicine, The University of Sheffield, Sheffield, UK. Grant sponsor: European Union’s Seventh Framework Programme (FP7/2007 – 2013) as part of the project VPH-DARE@IT (grant agreement no. 601055).
*Correspondence to: Deirdre McGrath, Ph.D., Centre for Computational Imaging & Simulation Technologies in Biomedicine, Department of Electronic and Electrical Engineering, The University of Sheffield, C14, Pam Liversidge Building, Mappin Street, Sheffield, S1 3JD, United Kingdom. Email:
[email protected] y These authors contributed equally to this study. Received 13 March 2015; revised 11 July 2015; accepted 19 July 2015 DOI 10.1002/mrm.25881 Published online 29 September 2015 in Wiley Online Library (wileyonlinelibrary.com). C 2015 Wiley Periodicals, Inc. V
Magnetic resonance elastography (MRE) enables noninvasive quantitative assessment of the mechanical properties of biological tissue (1). It involves three steps: 1) application of a mechanical wave [dynamic MRE (1)] or uniform compression [static or “quasi-static” MRE (2)] to biological tissue; 2) measurement of the displacements using MR imaging; 3) use of an inversion algorithm to calculate mechanical properties. Dynamic MRE, considered in this study, is being evaluated as a means of detecting changes in brain tissue mechanics associated with dementia and other neurodegenerative and neurological disorders (3–10). In fact, MRE has introduced the possibility of estimating brain tissue mechanics in vivo, as previously only in vitro measurement was feasible. Comparison between in vivo and ex vivo data is problematic, as biological tissue is subject to many alterations post excision, such as decay and loss of blood pressure. The MRE in vivo brain measurements obtained so far fall inside the wider range for ex vivo brain [e.g., shear stiffness is approximately 103104 Pa in vivo and 102105 ex vivo (11)]. However, MRE data for healthy brain varies widely (12–15), and understanding of this issue is critical to the development of MRE as a biomarker of neurodegenerative disease. Differing methodology between studies is a potential explanation. For instance, as brain tissue exhibits viscoelastic behavior (15), different MRE mechanical driving frequencies would elicit varying mechanical responses. However, even at similar frequencies there have been large inconsistencies. For instance, for healthy brain, Kruse et al. (14) reported a shear stiffness of 5.22 kPa at 100 Hz for gray matter and a much higher value of 13.6 kPa for white matter, whereas at a similar frequency of 90 Hz, Green et al. (16) measured much lower values, and moreover a lower stiffness of 2.7 kPa for white matter compared with 3.1 kPa for gray matter. Furthermore, the expected impact of neurodegenerative disease is relatively low. For example, Murphy et al. (3) reported only a 7% decrease in MRE-measured average brain shear stiffness for Alzheimer’s disease (i.e., 2.37 kPa for controls and 2.2 kPa for Alzheimer’s disease). For validation as a biomarker of
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neurodegeneration, MRE must be proven sufficiently accurate to detect these marginal variations. Although other methodological differences between MRE studies may underlie the lack of agreement in healthy data, it is important to consider the potential influence of the particular anatomy and geometry of the human brain, as well as their variation between individuals. There is growing evidence that cranial anatomy could strongly influence brain MRE. Clayton et al. (17) recently measured wave transmission and attenuation in brain MRE acquisitions and presented data suggesting that the processes of the meningeal dura mater, the falx cerebri and tentorium cerebelli membranes, can act as secondary wave sources through reflection and/or mechanical motion. Papazoglou et al. (18) proposed a compliance-weighted MRE method that exploits the effect of wave scattering through brain tissue types with differing elasticity to estimate the relative elasticity of subcortical regions. Their data from brain MRE acquisitions demonstrate large gradients of elasticity with the potential to cause wave scattering at the falx cerebri and the ventricles. Wave reflection, refraction and scattering can bring about distortions and interference patterns in the MRE displacement fields, which may introduce errors to the derived estimates of the biomechanical properties. Destructive interference of the reflected waves can cause areas of low wave amplitude with poor signal to noise (SNR), which pose difficulties for accurate MRE inversion (19), and some algorithms incorporate directional filters to counter this effect (20,21). It was therefore hypothesized that in brain MRE, waves would be reflected from the interfaces of brain tissue with cranial features, which would subsequently bring about distorted wave patterns, leading to errors in the calculated mechanical properties. Furthermore, it was postulated that these effects would vary across the brain volume as a function of the local geometry, and between individuals with different cranial shapes and volumes. It was also predicted that tissue heterogeneity at the brain boundaries with cranial features would affect the MRE inversions. To test this hypothesis, finite element (FE)-based simulations were performed on models with simplified twodimensional (2D) and three-dimensional (3D) geometries and on realistic 3D human brain models derived from anatomical MR imaging. To quantify the influence on MRE measurements, the brain mechanical properties were reconstructed from the simulated MRE displacement fields and compared with ground truth values. Previous studies have simulated dynamic MRE in the brain to evaluate inversion algorithms. Braun et al. (22) employed a coupled harmonic oscillator algorithm, and Murphy et al. (23) used the FE method to simulate 2D slices of human brain. Clayton et al. (24) simulated a 3D mouse brain, again using FE methods. To the best of our knowledge, the present study is the first attempt to simulate the full 3D wave field associated with dynamic MRE of the human brain, and to investigate thereby the patterns and effects of wave reflections from cranial features. THEORY Direct Inversion In dynamic MRE, several acquisitions are typically made to measure the displacement field of the propagating mechanical wave at different snapshots in time (16).
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These data are processed using Fourier transform to obtain the frequency domain complex displacement field describing the steady state: uðx; t Þ ¼ uðxÞexp ðivt Þ;
[1]
where v is the angular frequency of the input waveform. Brain tissue is commonly characterized using viscoelastic constitutive models (16,25). In near-incompressible brain tissue, the amplitude of the compressional component of the MRE wave is very small and difficult to measure accurately, but could cause large errors if ignored (26). A solution is to remove the contribution of the compressional component and to focus on estimating parameters of the shear component: the shear modulus (l) and the shear viscosity (f). This can be achieved by computing the curl (i.e., the divergence-free part) of the vector field u; obtaining a Helmholtz equation describing the steady state of the curl ðv ¼ r uÞ (26): rv2 v ¼ mr2 v þ ivz r2 v;
[2]
where q is the material density. In 3D, this is solved as three equations for the three components of v for two unknowns (l and f). This is the so-called “direct inversion” problem (16,19,27–29) and can be solved in matrix form using a least squares calculation (26). In 2D, the calculation reduces to simple division, as v has a single component. Alternative approaches include linear inversion (30), overlapping subzone method (31–34), and local frequency estimation (14,21). l and f are related to the complex shear modulus, G ðvÞ, which relates complex stress s ðvÞ to complex strain e ðvÞ: h 0 i 00 [3] s ðvÞ ¼ G ðvÞe ðvÞ ¼ G ðvÞ þ iG ðvÞ e ðvÞ; 0
00
where G ¼ m is the storage modulus and G ¼ vz the loss modulus. Acoustic Impedance Mismatch Acoustic waves are reflected at the interface of two tissue types according to the relative mismatch of acoustic impedances (z1 and z2), where Ai and Ar are the incident and reflected wave amplitudes, respectively: Ar 2 ¼ Ai 2
z2 z1 2 : z2 þ z1
[4]
The impedance for the shear wave component zs depends on the shear modulus and density of the material: zs ¼
pffiffiffiffiffiffi mr:
[5]
The compressional components may also mode convert to shear waves when reflected at the interfaces, again in proportion to the impedance mismatch. The compressional impedance, zp, also depends on the material properties: zp ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðl þ 2mÞ:
where l is Lame’s first parameter.
[6]
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Table 1 Constitutive Parameter Values Employed in Simulations Parameter Values Tissue Type Brain
0
Frequency (Hz) 25 37.5 50 62.5 90
G (Pa) 1110 1310 1520 2010 3100
G00 (Pa) 480 570 600 800 2500
E (MPa) FTM CSF
Poisson’s ratio 0.499
Poisson’s ratio
31.5 G0 (Pa) 1000
Density (kg/m3) 1000
0.45 G‘ (Pa) 900
METHODS Tissue Material Properties All material properties are listed in Table 1. Brain tissue was modeled as a soft homogeneous linear viscoelastic near-incompressible material with G0 and G00 as measured by MRE for healthy brain at 25–90 Hz (16,25). At 25–62.5 Hz, G0 and G00 for brain material were mean measurements for whole brain (25), and at 90 Hz, measurements for gray matter were employed (16). At each frequency, the whole brain volume (upper hemispheres and cerebellum) was set uniformly to the same properties to focus on sensitivity to cranial anatomy. For brain, the density was approximated to that of water (1000 kg/m3) (25) and Poisson’s ratio to that of a near-incompressible material (0.499). The falx cerebri, tentorium cerebelli, and falx cerebelli (FTM; Fig. 1e) were modeled as a linear elastic solid and cerebrospinal fluid (CSF) was modeled as a soft viscoelastic solid according the material specifications employed in previous brain simulations (35). Overview of Numerical Experiments All simulation details are summarized in Table 2. The MRE simulations were performed using Abaqus v6.12 (Dassault Systemes Simulia Corp, Johnston, Rhode Island, USA) direct-solution steady-state dynamic analysis. This algorithm is a perturbation procedure in which the response of a model to an applied harmonic vibration is calculated about a base state to produce frequency-space steady-state nodal displacements u, similar to Equation 1. All simulations were repeated for five frequencies (Table 1). Three types of model were employed: 2D squares, 3D cubes, and anthropomorphic brain shapes. The 2D square and 3D cube simulations allowed exploration of inversion accuracy in simplified geometries. The anthropomorphic models extended this assessment to the context of cranial geometry. When not specified otherwise, all other calculations were performed in MATLAB (R2012a, MathWorks Inc., Natick, Massachusetts, USA). Inversion Calculation and Assessment of Inversion Accuracy Direct inversion (solving Eq. 2) was implemented through derivative calculation using a finite difference
b (s21) 80
1130 K (MPa) 1050
1000
method on a “virtual imaging voxel” grid, which was sub-sampled (2D square and 3D cube) or interpolated (brain shape) at different intervals from the FE nodal displacements. To evaluate the accuracy of the inversions of G0 and G00 , the mean absolute percentage difference (MAPD) was calculated for the “virtual voxels” for a region or volume corresponding to brain tissue: N n 100 X Ggt Gi MAPD ¼ [7] ; N n Ggt where n is the voxel number, N is the total number of voxels in the volume being studied, Ggt is the ground truth value of the parameter (G0 or G00 ), and Gin is the inversion value of the parameter for voxel n. 2D Square Models Limiting the problem to 2D reduced computational overheads, facilitating the exploration of high resolution meshes. A square (10 10 cm2) of brain was meshed using quadratic eight-node quadrilateral plane strain elements (Fig. 1a). Convergence of the FE solution was confirmed by progressively refining the mesh from 4-mm element edge lengths down to 100 mm. The nearincompressible brain tissue was modeled with hybrid (linear pressure) elements to avoid volumetric locking, by discretizing and solving for the pressure field independently of the displacements. Loading consisted of xdirection complex harmonic displacement applied at one edge (Fig. 1a) with 30 mm amplitude (real component) [i.e., 30 mm is the approximate displacement amplitude at the brain edges in MRE (25)]. The y-direction displacements of the nodes on the loading edge were allowed to vary freely. The remaining boundary conditions (BCs) were varied between two options: 1) freely varying edge nodes at the nonloading edges (BC1) and 2) y-direction displacements fixed to zero at the nonloading edges (BC2) (Fig. 1a) to create a no-slip BC at the left and right edges and to reduce the displacements in the y-direction that emanated from these edges, thereby creating a simple wave pattern. Initial simulations compared BC1 and BC2, whereas later simulations included features to simulate the falx cerebri and a CSF layer on the nonloading edges (Fig. 1b) while applying BC2 to focus on
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FIG. 1. Meshes, BCs, loading conditions, and material types for MRE simulations. (a) 2D simulation BCs and loading. (b) 2D simulation material types. (c) 3D cube simulations material types and loading. (d) Brain shape whole volume mesh including submeshes for meningeal outer CSF layer, brain, ventricles, and the FTM membranes. (e) FTM submeshes. (f) Axial slice through brain mesh showing submeshes of brain tissue, falx cerebri, ventricles, meninges, and nodes for which displacements are applied for loading (trabeculae connection nodes [DNS1] and FTM edge nodes [yellow]). (g) Top panel: FTM edge nodes (red). Lower panel: DNS2 node set (red) consisting of the trabeculae connection nodes and FTM edge nodes.
reflections from the tissue interfaces by reducing reflections from the outer edges.
(Fig. 1c) and applied BC4 to focus on reflections from the brain–falx interface.
3D Cube Models
Anthropomorphic Brain Models
Cube simulations extended the problem to 3D in a limited fashion, with a simplified geometry. A cube (3 3 3 cm3) of brain was modeled and meshed with linear eight-node brick elements (hybrid constant pressure) at 0.5 mm element edge length. Loading consisted of 30 mm (real component) displacements to nodes on one face in a direction tangential to the face surface (z-direction) (Fig. 1c). For nodes on the loading face, free displacement was allowed in the x- and y-directions. For the nonloading faces, BCs were varied between two options: 1) nodes allowed to vary freely in all directions (BC3) and 2) fixing the displacements in the x- and ydirections to zero (BC4) to create a no-slip condition similar to BC2, thereby reducing the displacements in the xand y-directions emanating from these faces. Initial simulations compared BC3 and BC4, and the subsequent simulation included a feature to model the falx cerebri
Brain-shaped models were generated from MRI T1weighted acquisitions of healthy volunteers (n ¼ 5) from a public database (IXI database, brain-development.org; copyright: Imperial College of Science, Technology and Medicine, 2007). Meshes of multiple brain structures were generated using ISO2MESH, an open-source mesh generation and processing toolbox developed by Fang and Boas (36). This toolbox uses the surface and volumetric meshing algorithms available in CGAL (CGAL, Computational Geometry Algorithms Library, www.cgal. org) and Tetgen (wias-berlin.de/software/tetgen). Manual segmentations of each structure were merged to create a multilabel segmentation for each individual. Subsequently, the ISO2MESH “vol2mesh” function was invoked to generate linear tetrahedral meshes from the multilabel images, which treats each label as a distinct, closed surface and ensures the presence of shared nodes
Quadratic eight-node square hybrid linear pressure
Quadratic eight-node square, for brain hybrid linear pressure Quadratic eight-node square hybrid linear pressure Linear eight-node brick, hybrid constant pressure Linear eight-node brick, hybrid constant pressure
Linear eight-node brick, for brain hybrid constant pressure Linear four-node tetrahedral, for brain and CSF hybrid linear pressure
2D square (10 10 cm2)
2D square (10 10 cm2)
2D square (10 10 cm2)
3D cube (3 3 3 cm3)
3D cube (3 3 3 cm3)
3D cube (3 3 3 cm3)
Brain mesh #1
Brain mesh #1
Brain mesh #1
Brain mesh #1
10
11
12
13
14
15
16
17
18
19
Linear four-node tetrahedral, for brain and CSF hybrid linear pressure Linear four-node tetrahedral, for brain and CSF hybrid linear pressure Linear four-node tetrahedral, hybrid linear pressure
Quadratic eight-node square hybrid linear pressure
Element Type
2D square (10 10 cm2)
Model Shape
1–9
Simulation No.
Table 2 Details of Meshes, BCs, and Tissue Types Employed in Simulations No. of Elements
60 60 60 ¼ 216,000
592,284
771,465
1,240,192
1,077,188
1 mm
1 mm
1 mm
1 mm
60 60 60 ¼ 216,000
60 60 60 ¼ 216,000
800 800 ¼ 640,000
800 800 ¼ 640,000
1000 1000, 800 800, 600 600, 400 400, 300 300, 200 200, 100 100, 50 50, 25 25 800 800 ¼ 640,000
0.5
0.5
0.5
0.125
0.125
0.125
0.1, 0.125, 0.167, 0.25, 0.33, 0.5, 1, 2, 4
Element Edge Length (mm) Loading and BCs
DNS2
DNS2
30 mm x-direction displacement loading on one edge and BC2 (y-direction fixed on nonloading edges) 30 mm x-direction displacement loading on one edge and BC2 30 mm x-direction displacement loading on one edge and BC2 30 mm z-direction displacement on one face and BC3 (free nodes on nonloading faces) 30 mm z-direction displacement on one face and BC4 (x and y direction fixed on nonloading faces) 30 mm z-direction displacement on one face and BC4 30 mm head-foot displacement loading on cortical surface and FTM edge nodes (DNS2) DNS2
30 mm x-direction displacement loading on one edge and BC1 (free nodes on nonloading edges)
(Continued)
MT1: brain
MT4
MT4
MT4: brain, CSF, FTM
Brain and FTM
Brain
Brain
Brain and CSF
Brain and FTM
Brain
Brain
Tissue Types
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MT4 716,000–1,077,188 1 mm
30 mm head–foot displacement loading on cortical surface nodes only (DNS1)
MT4 716,000–1,077,188 1 mm
DNS2
716,000–1,077,188 1 mm
DNS2
No. of Elements 1,077,188 Model Shape Brain mesh #1
All brain meshes
All brain meshes
All brain meshes
Simulation No. 20
21–25
26–30
31–35
TABLE 2. Continued
Element Type Linear four-node tetrahedral, for brain hybrid linear pressure Linear four-node tetrahedral, hybrid linear pressure Linear four-node tetrahedral, for brain and CSF hybrid linear pressure Linear four-node tetrahedral, for brain and CSF hybrid linear pressure
Element Edge Length (mm) 1 mm
Loading and BCs DNS2
MT2: brain and CSF
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Tissue Types MT3: brain and FTM
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between adjacent structures in the final volumetric meshes (Fig. 1d). Element edge lengths were approximately 1 mm (limited by computational overheads). The models included submeshes for the whole brain, FTM, ventricles, and outer meningeal layer containing CSF (Fig. 1d–1f). Hybrid (linear-pressure) elements were used for submeshes with brain and CSF properties. The FTM membranes were approximately 1–2 mm in thickness, which is in agreement with literature values for healthy adult brain (37). Waves transmit from the skull to the brain via a combination of mechanical vibration of the layers of the meninges (dura mater, subarachnoid space, and pia mater) and the FTM and acoustic wave propagation through the meningeal CSF. The dura mater is tightly attached to the skull, whereas the pia mater is bound to the cortical surface, and the dura mater is joined to the pia mater via filaments running through the subarachnoid space called trabeculae. To simplify the problem of simulating wave delivery, loading was delivered at the brain submesh surface, which approximated the pia mater surface. Nodes were selected on this surface to model locations where the trabeculae connect to the pia mater from the dura mater, herein referred to as trabeculae connection nodes (Fig. 1f–1g). Harmonic displacement of 30 mm amplitude was applied to the selected nodes in the head–foot direction (i.e., the z-direction). These nodes were free to move in the x- and ydirections. Displacement loading rather than force loading was applied as it facilitated the control of the overall displacement amplitude, keeping it in line with values measured in literature for brain MRE, and was found to give rise to displacement fields that strongly resembled those observed in MRE acquisitions. Furthermore, the displacements (x, y, and z) of nodes on the outer surface of the meningeal CSF layer (dura mater–skull interface) were fixed to zero to limit the wave energy reflecting back into the brain to reduce the problem complexity. However, because the meninges are very effective at damping wave energy (17), it is likely that in reality very little wave energy would be reflected back into the brain. In some simulations, the outer edge nodes of the FTM that lay closest to the meninges (yellow nodes in Fig. 1f and red nodes in the top panel of Fig. 1g) were added to the displacement node set to simulate additional wave delivery via these structures (the trabeculae connection node set and the outer FTM edge node set were separate, with no nodes in common). In total, two displacement node sets (DNS) and four material type (MT) options were employed to explore various aspects. DNS1 comprised the trabeculae connection nodes alone, whereas DNS2 consisted of both the trabeculae connection nodes and the FTM edge nodes (Fig. 1f–1g). The nodes of DNS1 or DNS2 had a displacement loading applied in the z-direction while being free to move in the x- and ydirections. All other nodes were free to move in all directions (apart from the fixed outer dura mater nodes). The MT options were as follows: for MT1, all elements were set to brain material properties; for MT2, the material types included were brain and CSF (ventricles and the meningeal outer layer), whereas the FTM submesh was set to brain properties; for MT3, only brain and
Simulation of Influence of Cranial Anatomy in Brain MRE
FTM material types were modeled, whereas the CSF submeshes were set to brain properties; for MT4, all material types were included (brain, CSF, and FTM). For inversion, the nodal displacements were interpolated onto a “virtual voxel” grid using the MATLAB implementation of the “Natural Neighbor” interpolation algorithm (38). To reduce the influence of numerical inaccuracies in the FE solution and interpolation, the curl was smoothed using a 3 3 3 box filter prior to Laplacian calculation. Curl smoothing has also been performed on acquired brain MRE data (23) to reduce the influence of imaging-related noise. The small 3 3 3 filter dimension minimized averaging with surrounding tissue, and the box filter achieved better noise suppression than a 3 3 3 Gaussian filter. Convergence of the brain FE models was confirmed by comparing the interpolated displacement fields for different mesh resolutions for an example brain model (#1) with MT4 and DNS2. The original mesh had 1,077,188 elements. The model was remeshed with 592,284, 771,465, and 1,240,192 elements, and the interpolated displacement fields (3-mm voxel grid) of the lower resolution meshes were compared with that of the highest resolution mesh. Simulations were performed with brain model #1 (with 1,077,188 elements) and MT1 to determine the baseline inversion accuracy in the absence of reflective tissue interfaces. Interpolation stepsizes of 1–5 mm were explored, with and without curl smoothing. Following this, in order to understand the influences of wave delivery via FTM vibration and reflection from the various tissue interfaces, three sets of simulations were performed on all five brain meshes with MT2 þ DNS2, MT4 þ DNS2, and MT4 þ DNS1. Initially, the MAPD of G0 and G00 was calculated for a volume mask corresponding to brain tissue. Next, the MAPD values were recalculated for the mask eroded by a margin of three voxels to account for errors from partial volume with the FTM and CSF resulting from interpolation, smoothing of the curl, and the derivative calculation. The MAPD values from the full and eroded masks were compared to understand the sources of error on the inversions and the influence of cranial features.
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more rapidly varying curl at a shorter wavelength). Also for 25 Hz, when the stepsize was increased to 0.5 and 1 mm, the problem was offset by the larger step change in u. However, as the stepsize increased, artifacts appeared toward the edges, owing to the interference patterns resulting from reflections. Furthermore, as the stepsize increased, G0 was increasingly overestimated, whereas G00 was increasingly underestimated (but to a lesser extent). This was due to decreasing accuracy of the derivative estimates with increasing stepsize. With BC2, reflections at the left and right edges were reduced (Fig. 2c), and the associated artifacts were absent from G0 (Fig. 2d). Artifacts were also absent from G00 (data not shown). However, the errors associated with numerical precision remained at 25 Hz for 0.25 mm. Furthermore, the overestimation of G0 with increasing stepsize reoccurred. The MAPD (Fig. 2e, 2f) increased with stepsize but was greater for G0 than it was for G00 and for BC1 than BC2 and increased with frequency. In particular, with BC1 at 90 Hz for 4 mm the error was particularly large for G0 at 17%. As noted earlier, for BC2 at larger stepsizes, the error on G0 was negative and the error on G00 was positive, because G0 was overestimated and G00 was underestimated (see Eq. 7). Reflections at the brain–falx interface (uy and the divergence Fig. 3a) due to impedance mismatch resulted in inversion artifacts (Fig. 3b) that were accentuated at larger stepsizes and were accompanied by overestimation in G0 , especially at 90 Hz. Furthermore, a radial pattern of artifacts occurred at the tip of the falx. In addition, large artifacts and negative moduli values occurred at the brain–falx interface and were associated with material heterogeneity at this boundary, which violates the assumption of local homogeneity in the direct inversion. Similar reflections occurred at the brain–CSF interfaces (uy and divergence; Fig. 3c). However, in the inversions (Fig. 3d), much larger errors were noted at 90 Hz than at 25 Hz. This is likely to be influenced by a larger difference in the properties of brain and CSF at 90 Hz compared with 25 Hz. G0 was again overestimated at large stepsizes, especially at 90 Hz.
RESULTS Inversion Errors in 2D Simulations The 2D FE solution was observed to converge well with an element edge length of 125 mm (i.e., the root mean square error of displacements between the solutions for 125 mm and 100 mm was
